Frequency Stability of Graphene Nonlinear Parametric Oscillator

TL;DR

This work tackles the challenge of frequency stability in graphene nanomechanical resonators where strong nonlinearities limit performance. By operating graphene nanodrums as phase-locked parametric oscillators driven beyond the period-doubling regime, the authors demonstrate improved short-term frequency stability, quantified by a lower Allan deviation $\sigma_y(\tau)$ at fast integration times compared with Duffing oscillations at the same amplitude. A minimal theoretical model yields a phase-diffusion description with diffusion constant ${\rm D}_{\rm T-Par}(a_{ss}) = I_{\phi}(a_{ss}) + \left(\frac{3\gamma}{2\omega_0 \Gamma_{nl} a_{ss}}\right)^2 I_a(a_{ss})$, showing nonlinear damping suppresses amplitude-to-frequency noise conversion, while the Duffing case ${\rm D}_{\rm T-Duff} = I_{\phi}(a_{ss}) + \left(\frac{3\gamma a_{ss}^3 + 2S\cos\Delta}{4\omega_0 \Gamma_l a_{ss}^2}\right)^2 I_a(a_{ss})$ depends on the feedback phase. The results reveal phase-independent, amplitude-enhanced short-term stability in parametric operation, albeit with greater long-term drift due to actuator fluctuations. Overall, nonlinear dissipation emerges as a resource for high-precision graphene-based sensing, enabling fast, accurate oscillations in 2D-material devices.

Abstract

High-frequency stability is crucial for the performance of graphene resonators in sensing and timekeeping applications. However, the extreme miniaturization and high mechanical compliance that make graphene attractive also render it highly susceptible to nonlinearities, degrading frequency stability. Here, we demonstrate that graphene parametric oscillators provide an alternative nonlinear operating regime, where short-term frequency stability can be enhanced despite strong nonlinearity. By operating graphene resonators in a phase-locked loop (PLL), we experimentally demonstrate that parametric oscillations in the post-bifurcation regime achieve lower Allan deviation at fast integration times than Duffing oscillations at identical amplitudes. This improvement originates from strong nonlinear damping inherent to parametric oscillators, which suppresses amplitude-to-frequency noise conversion at large amplitudes. A minimal theoretical model captures observed phase diffusion and identifies nonlinear damping as the dominant mechanism governing phase noise reduction. These results highlight the role of nonlinear dissipation in enabling precision sensing beyond conventional limits of graphene oscillators.

Figures (4)

• Figure 1: a) Schematic of the experimental setup with a $14\,\mu\mathrm{m}$ diameter bilayer graphene nanodrum placed inside a vacuum chamber (10-4 mbar). The optical setup includes a blue laser for optothermal actuation and a red He-Ne laser for the readout. In the schematic, NF stands for neutral filter, PBS polarized beam splitter, CL convergent lens, BPF band-pass filter, PD photodetector, QWP quarter-wave plate, DM dichroic mirror, BS 92:8 pellicle beam splitter, and BS beam splitter. A lock-in amplifier is used for data acquisition and PLL measurements. b) Directly driven resonance of the graphene nanodrum. c) Parametric resonance of the graphene nanodrum. d) Phase measurement during and after PLL operation. PLL is activated for approximately 15 seconds and then deactivated. The phase-locking is shown in the inset. e) Allan deviation results of the closed-loop frequency measurements corresponding to the responses in (b) and (c). The PLL operation point for direct-linear response is taken at the peak amplitude (dashed line in (b)), and for the parametric response in the vicinity of the linear resonance frequency (dashed line in (c)).
• Figure 2: Phase-independent Allan deviation measurements of graphene parametric oscillator. The experiment is performed on a $12\,\mu\mathrm{m}$ diameter bilayer graphene nanodrum. a) Parametric closed-loop response curve for Vdrive = 2V. Three different PLL operation points are chosen close to the peak amplitude of the response curve (dashed lines), with nearly equal amplitudes but different phases. b) The Allan deviation results of the closed-loop measurements corresponding to the PLL operation points shown in a).
• Figure 3: Comparison between the frequency stability of Duffing and parametric oscillators. The experiments are performed for an $8\,\mu\mathrm{m}$ diameter multilayer graphene nanodrum. a) Frequency sweeps for the Duffing and parametric responses at the same drive level (80 mV). The horizontal line indicates a 17 mV response amplitude, while the color-matching dashed lines mark the PLL operating points. An exfoliated multilayer graphene nanodrum is used to achieve comparable response amplitudes for both Duffing and parametric cases, facilitating a direct comparison at the same drive level. b) Allan deviation results of the closed-loop frequency measurements corresponding to the responses shown in a). c) Theoretical frequency stability comparison calculated at a fixed integration time of 10 ms. The plot compares the stability of parametric and Duffing responses as a function of amplitude, assuming negligible nonlinear damping for the Duffing oscillator. The curves demonstrate that above a certain amplitude threshold, parametric stability is better than the Duffing and follows a similar trend to a linear oscillator, where no A-F noise conversion takes place.
• Figure 4: Drive amplitude vs peak amplitude relationships of direct and parametrically driven cases. a) Direct-driven frequency sweeps with Vdrive from 0.1 V to 2.4 V. These sweeps are taken using a $12\,\mu\mathrm{m}$ diameter bilayer graphene nanodrum. b) Drive amplitude vs. peak amplitude relationships of the direct-driven responses shown in a). The blue line represents a linear fit ($\mathrm{R^2} \approx 0.99$). c) Parametrically driven frequency sweeps with Vdrive from 0.4 V to 2.4 V. These sweeps are taken with a $12\,\mu\mathrm{m}$ diameter bilayer graphene nanodrum. d) Drive amplitude vs. peak amplitude relationships of the parametrically driven responses shown in c). The blue line represents a quadratic fit ($\mathrm{R^2} \approx 0.95$), supporting the model of non-negligible nonlinear damping. The dashed line shows a linear relation, which cannot exist in a bounded parametric response lifshitz2008nonlinear, highlighting the deviation from linearity in parametric oscillators.